Albert Einstein once told his son about his fame, “When a blind beetle crawls over the surface of a curved branch, it doesn’t notice that the track it has covered is indeed curved. I was lucky enough to notice what the beetle didn’t notice.”

By Thomas Levenson
Globe Correspondent
October 25, 2015

For Berlin’s weary citizens, November 1915 came with the bleak realization that times were not getting better any time soon. The First World War remained a bloody stalemate in trenches stretching from Switzerland to the North Sea. The British naval blockade was beginning to bite, and food shortages that would become desperate in 1916 were already being felt. Stomachs were going empty in one of Europe’s most learned, advanced cities, as the world around it was falling apart.

Albert Einstein had left his beloved Switzerland the year before, hoping to rub shoulders with the most impressive scientific minds in Europe. But he found most of his German colleagues to be war-crazed, including his closest companion, Fritz Haber, who turned poison gas into a weapon of war. Einstein couldn’t conceal his contempt, exhorting his fellows to “honor your master Jesus Christ, not only in words and songs, but rather foremost in your deeds.”

Still, wartime Berlin offered the increasingly isolated scientist the one thing he needed: time to think without interruption. For six weeks that fall, he seems to have done nothing else. And then, 100 years ago this November, Einstein completed his General Theory of Relativity — now recognized as the greatest individual scientific discovery of the last century.

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In his solitude, Einstein had been mulling a single, vexing problem: What happens when two (or more) objects — the sun and the earth, for example — exert a gravitational tug on each other? Until that point, there had been one generally accepted and elegantly simple answer in Isaac Newton’s law of gravitation. Published in 1687, it found that two bodies attract each other with a force that’s proportional to their combined mass and inversely proportional to the square of the distance between them.

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Newton’s law had worked nearly perfectly for more than 200 years. But cracks were emerging in the Newtonian facade. His theory posited that gravity acts instantaneously across empty space — the tug of the sun flashing out to grab the earth. Some of Newton’s contemporaries thought this an almost occult belief, close to blasphemy, but it didn’t seem to pose a fundamental problem until Einstein worked out the Special Theory of Relativity in 1905.

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Special relativity is based on the principle that nothing can travel faster than the speed of light — directly contradicting the Newtonian picture of immediate action-at-a-distance. There are more intricate ways of describing the conflict between Newtonian physics and the relativistic world view, but once Einstein called the legendary thinker into question, he felt compelled to sort it out.

Einstein’s first step toward a new description of gravity came with the realization that gravity and acceleration are two forms of the same experience. It’s impossible, he reasoned, for someone in a closed room to be sure whether the tug they feel toward the floor is that of the earth’s gravitational field pulling them down, or something — say a rocket — accelerating beneath, pushing upwards.

From there, Einstein started to analyze in detail what happens when you try to measure what happens during acceleration. Until he sat down to think about it, Einstein had held the traditional view that space and time form a four-dimensional structure, just the box the universe existed within. In this model, galaxies, stars, planets all move through the cosmos without any concern for the shape of space or the flow of time.

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What happens, Einstein wondered, to clocks in such a system — aboard a rocket ship, for example — accelerating out into the distance? Taking that scenario apart tick by tick he came to understand that time must pass more slowly — time bends — for someone riding the accelerating rocket, than it does for someone looking up from the pad at Cape Canaveral.

From there, the domino-like implications of his new line of thinking came into sharp relief. If acceleration bends time, so must gravity (they’re equivalent, after all). Because space and time are interwoven, gravity must then also warp the paths of objects traversing across distances as well as hours. The amount of bending, the local curvature of space-time, depends on how much matter and energy are present.

That warping of space-time must then affect the paths that matter-energy can take. That is: Space-time can’t be just the box that holds the cosmos. Instead, it is an active partner in a cosmic dance, affected by and affecting what it contains. As Einstein later put it, this was the crucial idea, that “the foundations of geometry have physical significance.”

That epiphany didn’t mean Einstein was done. But he’d taken the essential step. With it, he’d managed to take a pure physical insight — that gravity and acceleration are equivalent — and refine it to give a full conceptual account of what gravity might look like.

And here, Albert Einstein ran up against the limit of his own knowledge. He didn’t know enough math to prove what he so deeply felt to be true. But he did know a guy, his college friend turned mathematics professor, Marcel Grossman.

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Grossman held the key to moving beyond Euclidean geometry, which for more than two millennia had been as absolute an authority as there was in the scientific world. Until that point, no one had found an error in Euclid’s analyses of planes, surfaces, and solids. All of it seemed to be necessarily true — not only on the page, but in the real world.

Until it wasn’t. In the early and mid-19th century some of the most audacious thinkers in mathematics discovered they could modify one or another of Euclid’s axioms — those statements taken to be so obviously true that they required no proof. They built alternate geometries, just as consistent as Euclid’s, but ones in which, for example, no parallel lines exist. Grossman pointed Einstein to a version that could analyze how to make measurements at any point on a smoothly changing curved object — which is to say, exactly what Einstein now understood space-time would look like.

Most important, with this new mathematical formulation there was no longer any need for Newton’s occult, instant force-at-a-distance notion of gravity. Rather, what Newton had called a force is actually just the local curvature of space-time, the particular shape given to it by concentrations of mass-energy, like the earth or the sun. Objects navigating the cosmos — planets in orbit around a star, moons around a planet — simply follow the shortest route available to them around the dents and dips in space-time shaped by all the matter and energy in their vicinity.

Einstein’s gift for mental imagery showed itself when he tried to explain to his son how mere geometry could produce what we feel as the tug of gravity. Imagine, he said (at least so the story goes) a blind beetle. When it “crawls over the surface of a curved branch, it doesn’t notice that the track it has covered is indeed curved.”

Or imagine living on a vast, seemingly featureless plain, so flat that you know only two dimensions, length and width. Out for a walk one day, you find that your steps are coming harder. You begin to puff and labor. You sense that you’re being pulled by something — a force you could call gravity. It tugs you back as you walk along what you’re sure is a straight line. To anyone able to perceive three dimensions, not two, there is a simpler explanation — or as Einstein told his son, “I was lucky enough to notice what the beetle didn’t notice.”

That is: The “gravity” you feel on that plain is nothing more than the measure of a curvature of space, a rise you cannot actually see. The analogy isn’t perfect because it only deals in space, not time. But it gets to the nub of the matter: We inhabit a locally curved region of space and time created by the mass of the earth. The weight we feel as we stand by our beds in the morning is the sensation of our daily slide down a well in space-time, a warp bending down towards the center of the earth. That sensation, born of the geometry of experience, is an exercise in space-time dynamics that holds our feet to the floor.

Einstein.

That was as far as Einstein had gone by 1913. He was almost there conceptually, but he hadn’t yet mastered the new (to him) math that could express the interaction between matter and energy, space and time. There were still gaps in the theory, and at least one internal contradiction he couldn’t resolve. By the end of September 1915, he seemed ready to give up, confessing to a friend that “I do not believe that I myself am in the position to find the error, because my mind follows the same rut.”

Then, something happened. There’s no record of how he broke himself free, but over the first week of October 1915, he finally saw the way. For the next six weeks, he focused on gravity to the exclusion of everything else, until on the first Thursday in November, he was able to give his colleagues at the Prussian Academy the outlines of a near-complete theory.

Two weeks later, he returned with the first real confirmation that his seemingly absurd vision of torqued space and warped time was true. To put it to the test, he turned to a known anomaly in the solar system that Newtonian ideas had failed to explain — a tiny wobble in the orbit of Mercury, discovered in 1859.

Every prior explanation had failed — including the possibility that an unseen planet, dubbed Vulcan, in orbit very near the sun, was pulling Mercury off track. Now, Einstein simply plugged the numbers for the errant planet’s trajectory into his reworked equations, and looked at the answer. It was spot-on, confirmation, or at least strong evidence, that General Relativity could explain what Newton could not.

Einstein presented the final form of his theory on Nov. 25, 1915, a week after he ran the numbers on Mercury. Since then, General Relativity has been recognized as one of science’s biggest breakthroughs. That’s because it doesn’t simply reveal something new in the universe; it requires us to see the cosmos we inhabit in wholly new ways.

In Berlin, that last Thursday in November, it was hard to appreciate the full measure of the general theory’s transformative power. The war wouldn’t let Einstein — or anyone else — alone for long. Early in 1916, Karl Schwarzschild used general relativity to predict the possibility of what we now call black holes — his last finding before dying of an illness caught while on duty on the Eastern Front. Later that year, hunger turned to outright famine in the streets of Berlin.

And yet, amid catastrophe, Albert Einstein, simply by force of mind, managed to reconstruct a universe. The intense concentration on questions utterly removed from daily life offered him a refuge from the human folly and worse that surrounded him.

Years later, Einstein would try to describe what he felt over those last few weeks of solitude in a war-riven city, thinking about the universe. He couldn’t. “The years of searching in the dark for a truth that one feels but cannot express, the intense desire and the alternations of confidence and misgiving until one breaks through to clarity and understanding,” he wrote, “are known only to him who has experienced them.”

Thomas Levenson is a professor of science writing at MIT and an Ideas columnist. His book “The Hunt for Vulcan” will be published by Random House in November.